Chapter 1 Fluid Characteristics


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1 Chapter 1 Fluid Characteristics 1.1 Introduction Phases Solid increasing increasing spacing and intermolecular liquid latitude of cohesive Fluid gas (vapor) molecular force plasma motion Fluidity Fluid deform continuously under shearing (tangential) stresses no matter how small the stress stress time rate of angular deformation (strain, displacement) Newtonian fluid shear stress is linearly proportional to rate of angular deformation starting with zero stress and zero deformation constant of proportionality, namic viscosity Fig. 1.1 water, air [Cf] Analogy between Newtonian fluid and solids obeying Hooke's law of constant modulus of elasticity Solid deform by an amount proportional to the stress applied stress magnitude of the angular deformation (total strain) NonNewtonian fluid variable (nonlinear) proportionality between stress and deformation rate proportionality = f (length of time of exposure to stress, magnitude of stress) plastics: paint, jelly, polymer solutions Rheology 1 1
2 solid fluid Elastic Solid perfect memory Plastic partial memory Fluid zero memory 1 2
3 1.1.3 Compressibility 1) compressible fluid: gases, vapors thermonamics 2) incompressible fluid: liquid (small compressibility), water Continuum approach  dimensions in fluid space are large compared to the molecular spacing to ignore discrete molecular structure  neglect void  Consider a small volume of fluid V containing a large number of molecules, and let m and v be the mass and velocity of any individual molecule u lim V lim V m V vm m = volume which is sufficiently small compared with the smallest significant length scale in the flow field but is sufficiently large that it contains a large number of molecules [Cf] Molecular approach  molecular point of view  well developed for light gases 1 3
4 1 4 Ch 1. Fluid Characteristics
5 1.1.5 Noslip condition at rigid boundary 1) behavior of continuum  type viscous fluids 2) zero relative velocity at the boundary surface (proven by experiments) v 0 ideal fluid real fluid Multiphase system Single phase fluid combination of liquid gas combination of liquid vapor Multiphase systems cavitation problem combination of liquid solid combination of gas  solid sediment/pollutant transport 1.2 Units of Measurement SI system: metric system English system: ftlb system * Newton's 2nd law of motion F ma 1 5
6 F m a 2 force(n) ; mass(kg) ; acceleration(m / sec ) 2 F 1kgm /sec 1N W W mg weight ; g gravitational acceleration 1.3 Properties and States of Fluids 1) extensive properties ~ depend on amount of substance total volume, total energy, total weight 2) intensive properties ~ independent of the amount present volume per unit mass, energy per unit mass weight per unit volume (specific weight, γ ) pressure, viscosity, surface, tension Properties of importances in fluid namics (1) Pressure, p ~ scalar 2 p F / A (N/m ) p p p gauge absolute atm Forces on a fluid element Bo force: act without physical contact Surface force: require physical contact for transmission 1 6
7 1) bo force gravity force centrifugal force Coriolis force 2) surface forces normal stress tensile stress (unusual for fluid) pressure tangential stress shear stress (2) Temperature, T two bodies in thermal equilibrium same temperature (3) Density, = mass / volume = M V volume (pressure, temperature) (4) Specific weight, = weight / volume [Re] Flow of a continuous medium ~ Fluids are treated as homogeneous materials. ~ Molecular effects are disregarded. mass density ( xyzt,,, ) lim V 0 M V velocity vector s v lim t 0 t 1 7
8 (5) Viscosity, ~ due to molecular mobility ~ whenever a fluid moves such that a relative motion exists between adjacent volumes (different velocity) yx shear flow Stress, time rate of angular deformation i) displacement of AB relative to CD in t du du u y tut yt ii) strain = relative displacement = angular displacement du du yt / y t 1 8
9 iii) time rate of strain ( = time rate of angular displacement of AC) du t/ t du du yx du where yx shear stress acting in the x  direction on a plane whose normal is y  direction 2 (N / m ) du rate of angular deformation(1/ sec) namic molecular viscosity du ms m 2 N/m N s m 2 s m 2 (kg m / s ) kg / m sec kg/m s 2 Kinematic viscosity, kg / m s kg / m 2 m /s kinematic dimensions Fig
10 [Cf] namic: F, L, T kinematic: L, T shear stress deformation viscosity links two Types of Fluid Newtonian fluid constant and unique value of linear relation between and du NonNewtonian fluid ~ nonlinear du n Rheology, plastic 1 10
11 [Cf] Stressstrain relationship for solid yx d G d relative station displacement of AB d angular deformation ( shear strain) G = modulus of elasticity in torsion fluid solid velocity du d displacement = function of (temperature, pressure) major factor for viscosity when temperature is increasing Liquid intermolecular cohesion decrease cohesive force decrease viscosity Gas exchange of momentum increase molecular activity increase shear stress 1 11
12 [Re] Exchange of momentum fastspeed layer (FSL) molecules from FSL speed up molecules in LSL molecules from LSL slow down molecules in FSL Two layers tend to stick together as if there is some viscosity between two. lowspeed layer (LSL) Water: N 2 s m 1 12
13 decreases as T increases increases as T increases 1 13
14 (6) Specific heat, c = ratio of the quantity of heat flowing into a substance per unit mass to the change in temperature (7) Internal energy, u specific internal energy = energy per unit mass, Jkg kinetic + potential energy internal energy (8) Enthalpy specific enthalpy u p/ (9) Bulk modulus of elasticity and Compressibility 1) Compressibility, C = measure of change of volume and density when a substance is subjected to normal pressures or tensions = % change in volume (or density) for a given pressure change dvol C vol / dp d 1 dp 2) Bulk modulus of elasticity, E v E v 1 dp dp C dvol/ vol d / 1 14
15 p F / A dvol V1 V (10)Vapor pressure, pv = pressure at which liquids boil = equilibrium partial pressure which escaping liquid molecules will exert above any free surface ~ increases with temperature ~ The more volatile the liquid, the higher its vapor pressure. (11) Surface energy and surface tension, At boundaries between gas and liquid phase, molecular attraction introduce forces which cause the interface to behave like a membrane under tension. (force) (distance) work force area area length ~ water: decrease with temperature 1 15
16 Increase decrease then decrease decrease decrease increase 1 16
17 [Appendix 1] Coordinate Systems i) Cartesian ( x, yz, ) ii) Cylindrical ( R,, z) x Rcos y Rsin z z iii) Spherical (, r, ) x r sin cos y rsin sin z r cos x R cos R rsin 1 17
18 [Appendix 2] Tensor Scalar quantity with magnitude only Vector quantity with magnitude and direction Tensor an order array of entities which is invariant under coordinate transformation, this includes scalars and vectors Rank (order) of tensors 0th order 1 component, scalar (e.g., mass, length, pressure) 1st order 3 components, vector (e.g., velocity, force, acceleration) 2nd order 9 components, (e.g., stress, rate of strain, turbulent diffusion coeff.) Example of 2nd order tensor ~ stress acting on a fluid element Stress tensor xx xy xz yx yy yz zx zy zz = normal stress, = shear stress yx = shear stress in xz  plane and in x  direction 1 18
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